Simple Linear 2

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Equations:

$$\begin{align}\dot{x}_{0}&=x_{1}\\\dot{x}_{1}&=x_{0}\end{align}$$

Description:

A simple second order equation with constant coefficients.

Keywords:

simple; linear; constant coefficients

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Plots:

C5 WPD

Initial value problems:

24 (PILAU)
25 (PILANU)

IVP 24

Class: PILAU

Initial conditions:

$$\begin{align}x_{0}(0)&\subset~\left[ -1, 0.1\right]\\x_{1}(0)&\subset~\left[ 0.9, 1.1\right]\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&\subset~e^{t} \cdot \left[ -0.05, 0.6\right] + e^{- t} \cdot \left[ -1.05, -0.4\right]\\x_{1}(t)&\subset~e^{t} \cdot \left[ -0.05, 0.6\right] + e^{- t} \cdot \left[ 0.4, 1.05\right]\end{align}$$

View test data:

Solver
Settings
Results

VNODE-LP

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00400
c5: t_us=0.00210 , e_us=1.601337439971277e+00
c6: t_bd=10.000000 , e_bd=1.431720274620508e+04

VNODE-LP

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00800
c5: t_us=0.00430 , e_us=1.601337439971237e+00
c6: t_bd=10.000000 , e_bd=1.431720274619503e+04

VNODE-LP

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01190
c5: t_us=0.00720 , e_us=1.601337439971237e+00
c6: t_bd=10.000000 , e_bd=1.431720274619539e+04

VNODE-LP

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.02770
c5: t_us=0.02260 , e_us=1.601337439972813e+00
c6: t_bd=10.000000 , e_bd=1.431720274626275e+04

Generate graphics:

VNODE-LP

WPD for Problem 24

IVP 25

Class: PILANU

Initial conditions:

$$\begin{align}x_{0}(0)&=-1.0\\x_{1}(0)&=1.0\end{align}$$

Exact Solution^*:

$$\begin{align}x_{0}(t)&=- 1.0 \cdot e^{- t}\\x_{1}(t)&=1.0 \cdot e^{- t}\end{align}$$

View test data:

Solver
Settings
Results

VNODE-LP

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00400
c5: t_us=0.00300 , e_us=6.217248937900877e-15
c6: t_bd=10.000000 , e_bd=6.582230357418456e-11

VNODE-LP

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.00800
c5: t_us=0.00590 , e_us=6.106226635438361e-16
c6: t_bd=10.000000 , e_bd=1.158191795372137e-11

VNODE-LP

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01200
c5: t_us=0.00780 , e_us=4.440892098500626e-16
c6: t_bd=10.000000 , e_bd=1.015172771149861e-11

VNODE-LP

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.02640
c5: t_us=0.02170 , e_us=1.943445404606337e-13
c6: t_bd=10.000000 , e_bd=1.573298909934070e-09

Generate graphics:

VNODE-LP

WPD for Problem 25

^* rounded (non-rigorously) to 5 decimal places to preserve screen real estate.

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Simple Linear 2

Equations:

Description:

Keywords:

Plots:

Initial value problems:

IVP 24

Class: PILAU

Initial conditions:

Exact Solution*:

View test data:

Generate graphics:

IVP 25

Class: PILANU

Initial conditions:

Exact Solution*:

View test data:

Generate graphics:

Exact Solution^*:

Exact Solution^*: